Differential equations by direct integration
Problem 6.533 · easy
Solve \( \displaystyle y' = 2 x e^{x^{2}} \) with \( \displaystyle y(0) = -2 \).
- \[ \frac{d}{d x} e^{x^{2}} = 2 x e^{x^{2}} \]Integrate: y = this + C.✓ Proved
- \[ - \left. e^{x^{2}} \right|_{\substack{ x=0 }} - 2 = -3 \]y(0) = -2 fixes C.✓ Proved
- \[ \left. e^{x^{2}} - 3 \right|_{\substack{ x=0 }} = -2 \]The initial condition holds.✓ Proved
Answer \( y = e^{x^{2}} - 3 \)
Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | finite differences of the answer match f, and the initial value holds |
Reviewers
gpt-oss:20b: fail (misleading) — The solution incorrectly sets up the constant: it subtracts the value of exp(0) instead of adding it, yet still arrives at the correct constant. This misleads a student about how to apply the initial condition.qwen3.6:27b-mlx: pass — The solution correctly identifies the antiderivative, applies the initial condition to find the constant, and verifies the result. The reasoning is sound.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the antiderivative, applies the initial condition to find the constant, and verifies the result. The reasoning is sound.gpt-oss:20b: fail (misleading) 2026-10-18 — The solution incorrectly sets up the constant: it subtracts the value of exp(0) instead of adding it, yet still arrives at the correct constant. This misleads a student about how to apply the initial condition.qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the antiderivative, applies the initial condition to find the constant, and verifies the result. The logic is sound.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the antiderivative, applies the initial condition to find the constant, and verifies the result. The logic is sound.gpt-oss:20b: fail (misleading) 2026-10-14 — The solution incorrectly derives the constant C by writing "-Subs(exp(x**2), x, 0) - 2 = -3", which uses a negative of the exponential term and obscures the correct calculation 1 + C = -2 → C = -3. This could mislead a student about how to apply the initial condition.
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/de_direct_integration, checked 2026-10-18 with SymPy 1.14.0.